Many a times, while solving mathematical problems (specially proof ), you come across a situation where existence of some quantity pops up. Now, in such situation we try to prove the existence and stops there whereas we should proceed further to find out what that quantity or entity is ? We should ask ourselves can I find any method to figure out that quantity, because it may happen that method will turn into some algorithm. Other times we should look other properties of that entity i.e. does it satisfy all criteria of being a normal entity in the system or does it violate certain properties which gives you some new elements to define in that system.
"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos
"Don't just read it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? Where does the proof use the hypothesis?" - Paul Halmos